WASHINGTON — Senate supporters of far-reaching immigration legislation accepted minor changes in public while negotiating over more sweeping alterations in private Monday as they drove toward expected Judiciary Committee approval by mid-week.

In a long day of drafting, the panel voted to begin phasing in a requirement for foreigners to undergo fingerprinting when they leave the country. Lawmakers also agreed to make an immigrant’s third drunk driving conviction a deportable offense in some cases.

At the same time, officials expressed optimism that agreement was in sight in complex private talks over proposed changes to a section of the legislation relating to H-1B high skilled visas. As drafted, the bill would raise the current cap from 65,000 annually to 110,000, with the possibility of a further rise to 180,000.

Sen. Orrin Hatch (R-Utah) whose state is home to a burgeoning high tech industry, sought changes to reduce the cost and other conditions on firms that rely on highly skilled foreign labor.

He told reporters he is prepared to support the overall legislation when the committee votes on final passage if an agreement is reached on the issues. “The way it was written they’re going to move offshore,” he said of firms seeking changes.

In general, organized labor and its allies on the committee, including Sen. Richard Durbin (D-Ill.) wanted tougher conditions than industry was seeking, part of an attempt to assure that American workers are not disadvantaged by a larger influx of H-1B visa holders.

At its core, the legislation would provide an opportunity of U.S. citizenship to millions of immigrants living in the country illegally, create a new visa program for low-skilled workers and permit a sizeable increase in the number of high-tech visas, at the same time it mandates new measures to crack down on future unlawful immigration.

The full Senate is expected to begin debate on the legislation in June.

In two previous weeks of deliberations, supporters of the legislation have demonstrated their command over the committee’s proceedings, alternately accepting some proposals advanced by the bills critics and rejecting others — all without losing a single showdown.

The same pattern held true as the committee embarked on its third and final week of drafting.

On a vote of 13-5, the legislation’s supporters agreed to require foreigners leaving the country through any of the nation’s 30 busiest airports to submit to fingerprinting, part of an attempt to strengthen security.

“This is an agreement that we need to build toward a biometric visa exit system,” said Sen. Jeff Flake, R-Ariz. He said the system is “long overdue.” Most of the committee’s Democrats supported the provision, along with four Republicans. Among them were GOP Sens. Lindsey Graham of South Carolina and Flake of Arizona, two the so-called Gang of Eight that negotiated the bill’s basic framework over many months.

The committee last week rejected a proposal by Sen. Jeff Sessions, R-Ala., to impose the fingerprinting requirement at all of the nation’s ports of entry rather than only the biggest airports. He said the system’s partial implementation marked a “retreat from current law,” which already requires a nationwide biometric system. The requirement has not been fulfilled because of the cost.

Sen. Chuck Grassley, R-Iowa, advanced the drunk-driving amendment, a proposal that stipulates at least one of three convictions must occur after enactment of the law. It cleared on a vote of 17-1.

It appeared unlikely the issue of the high tech visas would be settled formally until Tuesday.

High-tech companies sought a change in the bill’s requirement that they show they’ve tried to recruit U.S. workers before hiring anyone on an H-1B visa. Hatch’s original proposal would make the regulation apply only on the firms most heavily dependent on H-1B visas, not on those where 85 percent of the jobs are filled by American citizens.

Hatch also has an amendment to change a requirement in the bill seeking to ensure that U.S. workers are not displaced by the hiring of foreigners.

In an email issued during the afternoon, the AFL-CIO urged its supporters to contact the committee and express opposition to “Hatch’s anti-worker amendments...

“Hatch’s amendments would change the bill so high-tech companies can hire new immigrant employees without first making the jobs available to American workers,” it said.

One Senate official said two earlier stabs at compromise had failed, one when the high tech industry rebelled at the terms, and the other when the AFL-CIO balked. The officials who described the deliberations did so on condition of anonymity, saying they were not authorized to speak on the record.

Whatever the ultimate outcome of the negotiations, it seemed unlikely that the committee’s vote would be the final vote on high tech visas — or any other major portion of the measure.

“This bill is not perfect,” Sen. Chuck Schumer, D-N.Y., said at one point during the day, one of several occasions in the past two weeks he has signaled that he and other supporters will be receptive to some changes when the legislation reaches the Senate floor.

And in fact, Sen. Marco Rubio, R-Fla., arguably the Republican with the largest ability to sway the vote in the Senate, has said he wants to tighten the fingerprinting requirement that the committee accepted during the day. “I will continue to fight to make the tracking of entries and exits include biometrics in the most effective system we can build when the bill is amended on the Senate floor,” he said.

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